Flanged Wheel Prolate Cycloid Problem: Train wheels have flanges that project beyond the

Chapter 13, Problem 7

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Flanged Wheel Prolate Cycloid Problem: Train wheels have flanges that project beyond the rims to keep the wheels from slipping off the track. A point P on the flange traces a prolate cycloid as the wheel turns. Figure 13-5k showsan example. The radius of the flange has been exaggerated so you can see more clearly what a prolate cycloid looks like. Assume the wheel has radius 50 cm, the flange has radius 70 cm, and that the wheel has rotated t radians since the point P(x, y) was farthest below the track. Let (not shown) be the position vector to point P(x, y) on the flange. Let be the vector from the origin to the point where the wheel touches the track. Let be the vector from that point to the center of the wheel. Let be the vector from the center of the wheel to the point on the flange. Figure 13-5k a. Explain why . b. The length of equals the distance the wheel has rolled. Vector is a constant vector in the vertical direction. Write and in terms of their components. c. Vector goes from the center of the wheel to point P(x, y). Write as a function of t. Use the result to write as a vector function of t. d. Plot the graph of using parametric mode. Does the graph look like Figure 13-5k? How far does P move in the x-direction between t = 0 radians and t = 0.1 radian? How do you explain the fact that the displacement is negative, even though the wheel is going in the positive x-direction?